Quadratic equations are fundamental in algebra and are widely used in various fields of science, engineering, and mathematics. They can be represented in the form:

a${x}^{2}$ + bx + c = 0

Where:

• x is the variable you want to solve for.
• a, b, and c are constants, with a ≠ 0.

• The quadratic formula is a universal method to solve any quadratic equation. It is given by:

x = (-b ± √(${b}^{2}$- 4ac)) / (2a)

• The "±" symbol means that there are usually two solutions: one with the plus sign and one with the minus sign. These solutions are often referred to as the "roots" of the equation.

• Use this formula to find the solutions (roots) of the quadratic equation.

b. Factoring: - If the quadratic equation can be factored into two binomials, it is easy to solve.

Example: If you have ${x}^{2}$ - 5x + 6 = 0, it can be factored as (x - 2)(x - 3) = 0.

c. Completing the Square: - Convert the quadratic equation into the form ${\left(x-p\right)}^{2}$ = q and then solve for x.

Example: For ${x}^{2}$ - 6x - 8 = 0, complete the square to get ${\left(x-3\right)}^{2}$ = 17.

2. Discriminant:

• The discriminant, D, is a value calculated from the coefficients of a quadratic equation: D = ${b}^{2}$ - 4ac. It provides information about the nature of the roots.

• If D > 0, the equation has two distinct real roots.

• If D = 0, the equation has one real root (a repeated root).

• If D < 0, the equation has two complex (non-real) roots.

3. Quadratic Equations in Real Life:

• Quadratic equations often represent physical phenomena, such as the motion of projectiles, the shape of parabolic mirrors, or financial modeling. They are also used in optimization problems.

• The vertex form of a quadratic equation is y = a${\left(x-h\right)}^{2}$ + k, where (h, k) is the vertex of the parabola.